Simplify and expand the following expression: $ \dfrac{2}{n - 6}+ \dfrac{4}{2n + 14}+ \dfrac{3n}{n^2 + n - 42} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the second term: $ \dfrac{4}{2n + 14} = \dfrac{4}{2(n + 7)}$ We can factor the quadratic in the third term: $ \dfrac{3n}{n^2 + n - 42} = \dfrac{3n}{(n - 6)(n + 7)}$ Now we have: $ \dfrac{2}{n - 6}+ \dfrac{4}{2(n + 7)}+ \dfrac{3n}{(n - 6)(n + 7)} $ The least common multiple of the denominators is: $ (n - 6)(n + 7)$ In order to get the first term over $(n - 6)(n + 7)$ , multiply by $\dfrac{2(n + 7)}{2(n + 7)}$ $ \dfrac{2}{n - 6} \times \dfrac{2(n + 7)}{2(n + 7)} = \dfrac{4(n + 7)}{(n - 6)(n + 7)} $ In order to get the second term over $(n - 6)(n + 7)$ , multiply by $\dfrac{n - 6}{n - 6}$ $ \dfrac{4}{2(n + 7)} \times \dfrac{n - 6}{n - 6} = \dfrac{4(n - 6)}{(n - 6)(n + 7)} $ In order to get the third term over $(n - 6)(n + 7)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{3n}{(n - 6)(n + 7)} \times \dfrac{2}{2} = \dfrac{6n}{(n - 6)(n + 7)} $ Now we have: $ \dfrac{4(n + 7)}{(n - 6)(n + 7)} + \dfrac{4(n - 6)}{(n - 6)(n + 7)} + \dfrac{6n}{(n - 6)(n + 7)} $ $ = \dfrac{ 4(n + 7) + 4(n - 6) + 6n} {(n - 6)(n + 7)} $ Expand: $ = \dfrac{4n + 28 + 4n - 24 + 6n}{2n^2 + 2n - 84} $ $ = \dfrac{14n + 4}{2n^2 + 2n - 84}$ Simplify: $ = \dfrac{7n + 2}{n^2 + n - 42}$